[Paper Review] Assouad-Nagata dimension of locally finite groups and asymptotic cones

J. Higes|arXiv (Cornell University)|Nov 9, 2007

Geometric and Algebraic Topology15 references3 citations

TL;DR

This paper resolves two open problems in geometric group theory concerning Assouad-Nagata dimension: it constructs locally finite groups with finite positive Assouad-Nagata dimension (answering negatively whether such dimension must be infinite), and provides metric spaces with positive Assouad-Nagata dimension whose asymptotic cones are all ultrametric (answering affirmatively a question about dimension behavior under asymptotic limits). The constructions use direct sums of finite groups with quasi-ultrametric structures and control of dilated cubes in product spaces.

ABSTRACT

In this work we study two problems about Assouad-Nagata dimension: 1) Is there a metric space of non zero Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes) 2) Suppose $G$ is a locally finite group with a proper left invariant metric $d_G$. If $\dim_{AN}(G, d_G)>0$, is $\dim_{AN} (G, d_G)$ infinite? (Brodskiy, Dydak and Lang) The first question is answered positively not only for general metric spaces but also for discrete groups with proper left invariant metrics. The second question has a negative solution. We show that for each $n$ there exists a locally finite group of Assouad-Nagata dimension $n$. A generalization to countable groups of arbitrary asymptotic dimension is given

Motivation & Objective

• To resolve Question 4.5 from [11], asking whether a metric space of positive Assouad-Nagata dimension can have all asymptotic cones of dimension zero.
• To answer Problem 5.3 from [6], which questions whether a locally finite group with positive Assouad-Nagata dimension must have infinite dimension.
• To construct explicit examples of countable locally finite groups with finite, arbitrary Assouad-Nagata dimension n ≥ 1.
• To demonstrate that asymptotic dimension and asymptotic Assouad-Nagata dimension can differ by a finite, arbitrary amount in countable groups.

Proposed method

• Constructs a locally finite group G^n as the direct sum of finite n-dimensional groups Z_{k_i}^n with increasing k_i.
• Equips the group with a quasi-ultrametric d_G generated from the word metrics on each factor, ensuring proper left-invariance.
• Uses dilated cubes in the product space to control the asymptotic Assouad-Nagata dimension via control functions of the form D(s) = C·s + k.
• Applies Lemma 4.7 and Lemma 4.8 to bound the diameter of s-scale connected components in covers, proving the dimension is exactly n.
• For infinite dimension, takes G = ⨁_{i=1}^∞ G_i^i with G_i^i = Z_{k_i}^i, showing asdim_AN(G) ≥ n for all n.
• Constructs a product group G^{(n,k)} = Z^n ⊕ G^k with a mixed metric to achieve asdim(G) = n and asdim_AN(G) = n+k.

Experimental results

Research questions

• RQ1 Is there a metric space of positive Assouad-Nagata dimension such that all its asymptotic cones are of dimension zero?
• RQ2 If a locally finite group has positive Assouad-Nagata dimension with respect to a proper left-invariant metric, must that dimension be infinite?
• RQ3 Can one construct a countable locally finite group with finite, arbitrary Assouad-Nagata dimension n ≥ 1?
• RQ4 Can the asymptotic dimension and asymptotic Assouad-Nagata dimension of a countable group differ by a finite, prescribed amount?
• RQ5 Does there exist a countable group of finite asymptotic dimension with finite but strictly larger asymptotic Assouad-Nagata dimension under some proper left-invariant metric?

Key findings

• For each n ∈ ℕ, there exists a locally finite group G^n with a proper left-invariant metric d_n such that asdim_AN(G^n, d_n) = n.
• The construction uses direct sums of finite n-dimensional groups Z_{k_i}^n with k_i increasing and r_i = floor(k_i/2), ensuring the existence of n-dimensional dilated cubes.
• The asymptotic cones of the constructed spaces are all ultrametric, answering Question 4.5 from [11] affirmatively.
• For each n and k with k ∈ ℕ ∪ {∞}, there exists a countable abelian group G^{(n,k)} with asdim(G^{(n,k)}, d_{(n,k)}) = n and asdim_AN(G^{(n,k)}, d_{(n,k)}) = n+k.
• The result shows that the difference between asymptotic dimension and asymptotic Assouad-Nagata dimension can be finite and arbitrarily large in countable groups.
• The paper disproves the possibility of defining the asymptotic Assouad-Nagata dimension of a group as the supremum of its finitely generated subgroups’ dimensions.